Players and coaching staff were "50 yards" from the Al Noor mosque in Christchurch, when the shooting began.
"If we were there five minutes earlier, it would have been worse," he added.
Prime Minister Jacinda Ardern said 49 people were killed in "terrorist attacks" at two mosques, while more than 40 people are seriously injured.
Mashud says the team "all are safe and sound" in the hotel but will travel home "in the coming days".
"Players were crying in the bus, they all were mentally affected," the former wicketkeeper told the BBC's Bengali Service.
The team arrived at the mosque on a bus following a news conference at the Hagley Oval - the venue of Saturday's now cancelled third Test. It is understood the news conference overran, leading to the delay in their arrival at the mosque.
"There were 17 members on the bus, as a manager I had the responsibility to return to the hotel with the boys. It's really hard, we fell like we were in a movie."
Earlier, some of the cricketers had described their ordeal on social media.
"Entire team got saved from active shooters," tweeted batsman Tamim Iqbal.
Wicketkeeper batsman Mushfiqur Rahim tweeted that the team was "extremely lucky" and he "never wants to see these things happen again".
Mohammad Isam, the Bangladesh correspondent for ESPN, told the BBC he was with the players at the time of the shooting.
"I saw them get out of the parking lot, within five minutes one of the players [Iqbal] called me for help - he said save us, we are in big trouble someone is shooting." said Isam.
"I didn't take him seriously at first but then his voice was cracking up and I just ran for it. I tried to run all the way and I got a lift from someone and I reached the incident.
"I tried to charge towards the team bus, which I saw from about 100 yards, I thought just go near to what was happening, there was live shooting going on at that time, there was fire - I saw one dead body and one person running towards me with a bloodied shoulder.
"By the time I got close to the park, the players had disembarked from the bus, they were running towards me and just telling me to get out of there.
"We ran through the park and headed back to the ground for safety and were there for about an hour."
He added: "The players were breaking down, they had seen way too much in the 15 minutes they were held up in the bus, there was no security because it is such a peaceful country."
"The players heard shots being fired, they saw people tumbling out of the gates and ducked under the bus."
"We've cancelled the game," said New Zealand Cricket chief executive David White. "I've spoken to my counterpart at Bangladesh cricket - we agree it's inappropriate to play cricket at this time.
"Both teams are deeply affected. As a country, we'll have to look at [security of visiting teams]. It seemed to be a safe haven. I'm sure all of New Zealand will take a look at their approach."
The International Cricket Council says it "fully supports" the decision to cancel the match.
The attacks happened at Al Noor mosque, located in central Christchurch and another in the suburb of Linwood.
A male in his late 20s has been charged with murder, while three others have been detained.
Williams leads tributes
New Zealand Rugby World Cup winner Sonny Bill Williams says he is "deeply saddened" by the attacks.
The 33-year-old, who converted to Islam in 2009, recorded an emotional message on social media.
"I'm just deeply, deeply saddened that this would happen in New Zealand," he said.
Former All Blacks and Wales player Shane Howarth said it was " a very sad day".
"To the Muslim community, I can only say sorry and that you are welcome in my country."
The the attackers, he added: "You have stained and scarred our country, you do not represent me or my values and I hope you rot in hell."
Pakistan president Imran Khan, a former international cricketer, tweeted: "Shocked and strongly condemn the Christchurch, New Zealand terrorist attack on mosques.
"This reaffirms what we have always maintained: that terrorism does not have a religion. Prayers go to the victims and their families."
While Ireland's New Zealand-born centre Bundee Aki, tweeted: "My prayers goes out to all the families, friends affected by this tragic news."